Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations

Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 796851, 13 pages doi:10.1155/2008/796851 Research Article Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations Ravi P. Agarwal,1Victoria Otero-Espinar,2Kanishka Perera,1 and Dolores R. Vivero2 1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA 2Departamento de An´ alise Matem´ atica, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain Correspondence should be addressed to Ravi P. Agarwal, [email protected] Received 21 April 2008; Accepted 17 August 2008 Recommended by Paul Eloe This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation. Copyright q2008 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The Emden-Fowler equation, uΔΔtqtuασt0,t∈0,1T,1.1 arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting system see, e.g., 1and the references thereinfor the continuous model. The negative exponent Emden-Fowler equation α<0 has been used in modeling non-Newtonian fluids such as coal slurries 2. The physical interest lies in the existence of positive solutions. We are interested in a broad class of singular problem that includes those related with 1.1and the more general equation uΔΔtqtuασtgt, uσt,t∈0,1T.1.2 Recently, existence theory for positive solutions of second-order boundary value problems on time scales has received much attention see, e.g., 3–6for general case, 7 for the continuous case, and 8for the discrete case. 2 Advances in Difference Equations In this paper, we consider the second-order dynamic equation with homogeneous Dirichlet boundary conditions: P⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −uΔΔtFt, uσt,Δ-a.e.t∈Dκo, ut>0,t∈a, bT, ua0ub, 1.3 where we say that a property holds for Δ-a.e. t∈A⊂Tor Δ-a.e. on A⊂T,Δ-a.e., whenever there exists a set E⊂Awith null Lebesgue Δ-measure such that this property holds for every t∈A\E,Tis an arbitrary time scale, subindex Tmeans intersection to T,a, b ∈Tare such that a<ρb,Da, bT,Dκa, ρbT,Dκ2a, ρ2bT,Doa, bT,Dκoa, ρbT, and F:D×0,∞→Ris an L1 Δ-Carath´ eodory function on compact subintervals of 0,∞, that is, it satisfies the following conditions. CiFor every x∈0,∞,F·,xis Δ-measurable in Do. iiFor Δ-a.e. t∈Do,Ft, ·∈C0,∞. CcFor every x1,x 2∈0,∞with x1≤x2, there exists mx1,x2∈L1 ΔDosuch that Ft, x≤mx1,x2tfor Δ-a.e.t∈Do,x∈x1,x 2.1.4 Moreover, in order to use variational techniques and critical point theory, we will assume that Fsatisfy the following condition. PMFor every x∈0,∞, function PF:D×0,∞→Rdefined for Δ-a.e. t∈Dand all x∈0,∞,as PFt, x:x 0 Ft, rdr, 1.5 satisfies that PF·,xis Δ-measurable in Do. We consider the spaces C1 0,rdDκ:C1 rdDκ∩C0D, C1 c,rdDκ:C1 rdDκ∩CcD, 1.6 where C1 rdDκis the set of all continuous functions on Dsuch that they are Δ-differentiable on Dκand their Δ-derivatives are rd-continuous on Dκ,C0Dis the set of all continuous functions on Dthat vanish on the boundary of D,andCcDis the set of all continuous functions on Dwith compact support on a, bT. We denote as ·CDthe norm in CD,that is, the supremum norm. On the other hand, we consider the first-order Sobolev spaces H1 ΔD:v:D−→ R:v∈ACD,v Δ∈L2 ΔDo, HH1 0,ΔD:v:D−→ R:v∈H1 ΔD,va0vb, 1.7 Ravi P. Agarwal et al. 3 where ACDis the set of all absolutely continuous functions on D. We denote as t2 t1 fsΔst1,t2T fsΔsfor t1,t 2∈D, t1<t 2,f∈L1 Δt1,t 2T.1.8 The set His endowed with the structure of Hilbert space together with the inner product ·,·H:H×H→Rgiven for every v, w∈H×Hby v,wH:vΔ,wΔL2 Δ:b a vΔs·wΔsΔs;1.9 we denote as ·Hits induced norm. Moreover, we consider the sets H0,loc :H1 loc,ΔD∩C0D, Hc,loc :H1 loc,ΔD∩CcD, 1.10 where H1 loc,ΔDis the set of all functions such that their restriction to every closed subinterval Jof a, bTbelong to the Sobolev space H1 ΔJ. We refer the reader to 9–11for an introduction to several properties of Sobolev spaces and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to 12for a broad introduction to dynamic equations on time scales. Definition 1.1. uis said to be a solution in the sense of distributions to Pif u∈H0,loc,u>0 on a, bT, and equality b auΔs·ϕΔs−Fs, uσs·ϕσsΔs01.11 holds for all ϕ∈C1 c,rdDκ. From the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2, we deduce that if uis solution in the sense of distributions, then, 1.11holds for all ϕ∈Hc,loc. This paper is devoted to prove the existence of multiple positive solutions to Pby using perturbation and variational methods. This paper is organized as follows. In Section 2, we deduce sufficient conditions for the existence of solutions in the sense of distributions to P. Under certain hypotheses, we approximate solutions in the sense of distributions to problem Pby a sequence of weak solutions to weak problems. In Section 3, we derive some sufficient conditions for the existence of at least one or two positive solutions to P. These results generalize those given in 7for T0,1, where problem Pis defined on the whole interval 0,1∩Tand the authors assume that F∈C0,1×0,∞,Rinstead of Cand PM.Thesufficient conditions for the existence of multiple positive solutions obtained in this paper are applied to a great class of bounded time scales such as finite union of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets among others. 4 Advances in Difference Equations 2. Approximation to Pby weak problems In this section, we will deduce sufficient conditions for the existence of solutions in the sense of distributions to P, where Ffgand f,g :D×0,∞→Rsatisfy Cand PM,f satisfies Cc,andgsatisfies the following condition. CgFor every p∈0,∞, there exists Mp∈L1 ΔDosuch that gt, x≤Mptfor Δ-a.e.t∈Do,x∈0,p.2.1 Under these hypotheses, we will be able to approximate solutions in the sense of distributions to problem Pby a sequence of weak solutions to weak problems. First of all, we enunciate a useful property of absolutely continuous functions on Dwhose proof we omit because of its simplicity. Lemma 2.1. If v∈ACD,thenv±:max{± v,0}∈ACD, vΔ−vΔ·vΔ≤0,v−ΔvΔ·v−Δ≤0,2.2 Δ-a.e. on Do. We fix {εj}j≥1a sequence of positive numbers strictly decreasing to zero; for every j≥1, we define fj:D×0,∞→Ras fjt, xft, max x, εj for every t, x∈D×0,∞.2.3 Note that fjsatisfies Cand Cg; consider the following modified weak problem Pj⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −uΔΔtfjt, uσtgt, uσt,Δ-a.e.t∈Dκo, ut>0,t∈a, bT, ua0ub. 2.4 Definition 2.2. uis said to be a weak solution to Pjif u∈H,u>0ona, bT, and equality b auΔs·ϕΔs−fjs, uσsgs, uσs·ϕσsΔs02.5 holds for all ϕ∈C1 0,rdDκ. uis said to be a weak lower solution to Pjif u∈Hu>0ona, bT, and inequality b auΔs·ϕΔs−fjs, uσsgs, uσs·ϕσsΔs≤02.6 holds for all ϕ∈C1 0,rdDκsuch that ϕ≥0onD. Ravi P. Agarwal et al. 5 The concept of weak upper solution to Pjis defined by reversing the previous inequality. We remark that the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2allows to assert that relations in Definition 2.2 are valid for all ϕ∈Hand for all ϕ∈Hsuch that ϕ≥0onD, respectively. By standard arguments, we can prove the following result. Proposition 2.3. Assume that f,g :D×0,∞→Rsatisfy (Cand (PM,fsatisfies (Cc, and gsatisfies (Cg. Then,ifforsomej≥1there exist ujand ujas a lower and an upper weak solution, respectively, to Pjsuch that uj≤ujon D,thenPjhas a weak solution uj∈uj, uj:{v∈H:uj≤v≤ ujon D}. Next, we will deduce the existence of one solution in the sense of distributions to Pfrom the existence of a sequence of weak solutions to Pj. In order to do this, we fix {ak}k≥1,{bk}k≥1⊂Dtwo sequences such that {ak}k≥1⊂a, ab/2Tis strictly decreasing to aif aσa,akafor all k≥1ifa<σaand {bk}k≥1⊂ab/2,bTis strictly increasing to bif ρbb,bkbfor all k≥1ifρb<b. We denote that Dk:ak,b kT, k≥1. Moreover, we fix {δk}k≥1a sequence of positive numbers strictly decreasing to zero such that σak,ρbkT⊂aδk,b−δkT,δ k≤b−a 2for k≥1.2.7 Proposition 2.4. Suppose that Ffgand f,g :D×0,∞→Rsatisfy (Cand (PM,f satisfies (Cc, and gsatisfies (Cg. Then, if for every j≥1,uj∈His a weak solution to Pjand νδ:inf j≥1min aδ,b−δT ,u j>0∀δ∈0,b−a 2,2.8 M:sup j≥1 max Duj<∞,2.9 then a subsequence of {uj}j≥1converges pointwise in Dto a solution in the sense of distributions u1 to P. Proof. Let k≥1 be arbitrary; we deduce, from 2.2,2.7,2.8,and2.9, that there exists a constant Kk≥0 such that for all j≥1, bk akuΔ js2ΔsuΔ jak2·μakuΔ jρbk2·μρbk ρbk σak uΔ js·uj−νδkΔsΔs ≤Kkuj,uj−νδkH. 2.10 6 Advances in Difference Equations Therefore, for all j≥1 so large that εj<ν δ1,asujis a weak solution to Pj, by taking ϕ1:uj−νδ1∈Has the test function in 2.5,from2.9,Ccand Cg, we can assert that there exists l∈L1 ΔDosuch that b1 a1uΔ js2Δs≤K1b a Fs, uσ js·ϕσ 1sΔs ≤K1Mb a lsΔs, 2.11 that is, {uj}j≥1is bounded in H1 ΔD1and hence, there exists a subsequence {u1j}j≥1which converges weakly in H1 ΔD1and strongly in CD1to some u1∈H1 ΔD1. For every k≥1, by considering for each j≥1 the weak solution to Pkjukjand by repeating the previous construction, we obtain a sequence {uk1j}j≥1which converges weakly in H1 ΔDk1and strongly in CDk1to some uk1∈H1 ΔDk1with {uk1j}j≥1⊂ {ukj}j≥1. By definition, we know that for all k≥1, uk1|Dkuk. Let u1:D→Rbe given by u1:ukon Dkfor all k≥1andu1a:0:u1bso that u1>0ona, bT,u1∈H1 loc,ΔD∩Ca, bT,u1is continuous in every isolated point of the boundary of D,and{ukk}k≥1converges pointwise in Dto u1. We will show that u1∈C0D; we only have to prove that u1is continuous in every dense point of the boundary of D.Let0<ε<Mbe arbitrary, it follows from Ccand Cg that there exist mε∈L1 ΔDosuch that mε≥0onDoand Ft, x≤mεtfor Δ-a.e. t∈Doand all x∈ε,M;letϕε∈Hbe the weak solution to −ϕΔΔ εtmεt,Δ-a.e.t∈Dκo,ϕ εa0ϕεb;2.12 we know see 4 that ϕε>0ona, bT. For all k≥1 so large that εkk<ε,sinceukkand ϕεare weak solutions to some problems, by taking ϕ2ukk−ε−ϕε∈Has the test function in their respective problems, we obtain ukk,ϕ2Hb a Fs, uσ kks·ϕσ 2sΔs ≤b a mεs·ϕσ 2sΔsϕε,ϕ2H; 2.13 thus, 2.2yields to  ϕ2 2 H≤ukk−ϕε,ϕ2H≤0,2.14 which implies that 0 ≤ukk≤εϕεon Dand so 0 ≤u1≤εϕεon D. Thereby, the continuity of ϕεin every dense point of the boundary of Dand the arbitrariness of εguarantee that u1∈C0D. Finally, we will see that 1.11holds for every test function ϕ∈C1 c,rdDκ;fixoneof them. For all k≥1 so large that supp ϕ⊂ak,b kTand all j≥1 so large that εkj<ν δk,asukj is a weak solution to Pkj, by taking ϕ∈C1 c,rdDκ⊂C1 0,rdDκas the test function in 2.5 and bearing in mind 2.7, we have bk ak uΔ kjs·ϕΔsΔsukj,ϕ Hbk ak Fs, uσ kjs·ϕσsΔs, 2.15 Ravi P. Agarwal et al. 7 whence it follows, by taking limits, that bk akukΔs·ϕΔs−Fs, ukσs·ϕσsΔs0,2.16 which is equivalent because u1|Dkukand ϕ0ϕσon Do\Do kto b auΔ 1s·ϕΔs−Fs, uσ 1s·ϕσsΔs0,2.17 and the proof is therefore complete. Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of at least one solution in the sense of distributions to problem P. Corollary 2.5. Let Ffgbe such that f,g :D×0,∞→Rsatisfy (Cand (PM,fsatisfies (Cc,and gsatisfies (Cg. Then, if for each j≥1there exist ujand uja lower and an upper weak solution, respectively, to Pjsuch that uj≤ujon Dand inf j≥1min aδ, b−δT uj>0∀δ∈0,b−a 2,sup j≥1 max Duj<∞,2.18 then Phas a solution in the sense of distributions u1. Finally, fixed u1∈H0,locis a solution in the sense of distributions to Pwith Ffg, we will derive the existence of a second solution in the sense of distributions to Pgreater than or equal to u1on D. For every k≥1, consider the weak problem  Pk⎧ ⎨ ⎩ −vΔΔtFt, u1vσt−Ft, uσ 1t,Δ-a.e.t∈Dκ ko, vak0vbk. 2.19 For every k≥1, consider Hk:H1 0,ΔDkas a subspace of Hby defining it for every v∈Hkas v0onD\Dkand define the functional Φk:Hk⊂H→Rfor every v∈Hkas Φkv:1 2 v 2 H−bk ak Gs, vσsΔs, 2.20 where function G:D×0,∞→Ris defined for Δ-a.e. t∈Dand all x∈0,∞as Gt, x:x 0Ft, uσ 1tr−Ft, uσ 1tdr. 2.21 As a consequence of Lemma 2.1, we deduce that every weak solution to  Pkis nonnegative on Dkand by reasoning as in 4,Section3, one can prove that Φkis weakly lower semicontinuous, Φkis continuously differentiable in Hk, for every v,w ∈Hk, Φ kvwv,wH−bk akFs, u1vσs−Fs, uσ 1s·wσsΔs, 2.22 and weak solutions to  Pkmatch up to the critical points of Φk. Next, we will assume the following condition. NIFor Δ-a.e. t∈Do,ft, ·is nonincreasing on 0,∞. 8 Advances in Difference Equations Proposition 2.6. Suppose that Ffgis such that f,g :D×0,∞→Rsatisfy (Cand (PM, fsatisfies (Ccand (NI, and gsatisfies (Cg. If {vk}k≥1⊂H,vk∈Hkis a bounded sequence in Hsuch that inf k≥1Φkvk>0,lim k→∞ Φ kvk H∗ k 0,2.23 then {vk}k≥1has a subsequence convergent pointwise in Dto a nontrivial function v∈Hsuch that v≥0in Dand u2:u1vis a solution in the sense of distributions to P. Proof. Since {vk}k≥1is bounded in H, it has a subsequence which converges weakly in Hand strongly in C0Dto some v∈H. For every k≥1, by 2.2,weobtain  v− k H≤ Φ kvk H∗ k,2.24 which implies, from 2.23,thatv≥0onDand so u2:u1v>0ona, bT. In order to show that u2:u1v∈H0,loc is a solution in the sense of distributions to P,fixϕ∈C1 c,rdDkarbitrary and choose k≥1 so large that supp ϕ⊂ak,b kT, bearing in mind that u1is a solution in the sense of distributions to P, and the pass to the limit in 2.22with vvkand wϕyields to 0b avΔs·ϕΔs−Fs, u1vσs−Fs, uσ 1s·ϕσsΔs b auΔ 2s·ϕΔs−Fs, uσ 2s·ϕσsΔs; 2.25 thus, u2is a solution in the sense of distributions to P. Finally, we will see that vis not the trivial function; suppose that v0onD. Condition NIensures that function Gdefined in 2.21satisfies for every k≥1andΔ-a.e. s∈Do, Gs, v kσs≥fs, u1v kσs−fs, uσ 1s·v kσs v kσs 0gs, uσ 1sr−gs, uσ 1sdr, 2.26 so that, by 2.20and 2.22, we have, for every k≥1, Φkvk≤1 2 vk 2 H−vk,v kHΦ  kvkv k −b ags, u1v kσs−gs, uσ 1s·v kσsΔs b av kσs 0gs, uσ 1sr−gs, uσ 1sdrΔs; 2.27 moreover, as we know that v k≤pon Dfor some p>0, it follows from Cgthat there exists m∈L1 ΔDosuch that Φkvk≤1 2 v− k 2 H− v k 2 HΦ  kvkv k2b a ms·v kσsΔs ≤1 2 v− k 2 H Φ kvk H∗ k · v k H2b a ms·v kσsΔs, 2.28 Ravi P. Agarwal et al. 9 and hence, since {v k}k≥1is bounded in Hand converges pointwise in Dto the trivial function v, we deduce, from the second relation in 2.23and 2.24, that limk→∞Φkvk≤0 which contradicts the first relation in 2.23. Therefore, vis a nontrivial function. 3. Results on the existence and uniqueness of solutions In this section, we will derive the existence of solutions in the sense of distributions to P where Ffg0ηg1,η≥0 is a small parameter, and f, g0,g 1:D×0,∞→Rsatisfy C, PMas well as the following conditions. H1There exists a constant x0∈0,∞and a nontrivial function f0∈L1 ΔDosuch that f0≥0Δ-a.e. on Doand ft, x≥f0t,g 0t, x,g 1t, x≥0forΔ-a.e.t∈Do,x∈0,x 0.3.1 H2For every p∈0,∞, there exist mp∈L1 ΔDoand Kp≥0 such that ft, x≤mptfor Δ-a.e.t∈Do,x∈p, ∞, g1t, x≤Kpfor Δ-a.e.t∈Do,x∈0,p. 3.2 H3There are m0∈L2 ΔDosuch that g0t, x≤λx m0tfor Δ-a.e.t∈Do,x∈0,∞,3.3 for some λ<λ 1, where λ1is the smallest positive eigenvalue of problem −uΔΔtλuσt,t∈Dκ2, ua0ub. 3.4 3.1. Existence of one solution. Uniqueness Theorem 3.1. Suppose that f,g0,g 1:D×0,∞→Rsatisfy (C,(PM,and (H1–H3. Then, there exists a η0>0such that for every η∈0,η 0, problem Pwith Ffg0ηg1has a solution in the sense of distributions u1. Proof. Let η≥0 be arbitrary; conditions H1–H3guarantee that g:g0ηg1satisfies Cg. We will show that there exists a η0>0 such that for every η∈0,η 0, hypotheses in Corollary 2.5 are satisfied. Let x0and f0be given in H1,weknow,from4, Proposition 2.7, that we can choose ε∈0,1so small that the weak solution u∈Hto −uΔΔtεf0t,Δ-a.e.t∈Dκo,ua0ub,3.5 satisfies that u>0ona, bTand u≤x0on D. Let j≥1 be so large that εj<x 0, we obtain, by H1,that −uΔΔt≤f0t≤fjt, uσtgt, uσt,Δ-a.e.t∈Do,3.6 whence it follows that uis a weak lower solution to Pj.